Density-functional expansion methods: evaluation of LDA, GGA, and meta-GGA functionals and different integral approximations

We extend the Kohn-Sham potential energy expansion (VE) to include variations of the kinetic energy density and use the VE formulation with a 6-31G* basis to perform a "Jacob's ladder" comparison of small molecule properties using density functionals classified as being either LDA, GGA, or meta-GGA. We show that the VE reproduces standard Kohn-Sham DFT results well if all integrals are performed without further approximation, and there is no substantial improvement in using meta-GGA functionals relative to GGA functionals. The advantages of using GGA versus LDA functionals becomes apparent when modeling hydrogen bonds. We furthermore examine the effect of using integral approximations to compute the zeroth-order energy and first-order matrix elements, and the results suggest that the origin of the short-range repulsive potential within self-consistent charge density-functional tight-binding methods mainly arises from the approximations made to the first-order matrix elements.     

Is it worthwhile to go beyond the local-density approximation in subsystem density functional theory?

Frozen density embedding (FDE) theory is one of the major techniques aiming to bring modeling of extended chemical systems into the realm of high accuracy calculations. To improve its accuracy it is of interest to develop kinetic energy density functional approximations specifically for FDE applications. In the study reported here we focused on optimizing parameters of a generalized gradient approximation-like kinetic energy functional with the purpose of better describing electron excitation energies. We found that our optimized parametrizations, named excPBE and excPBE-3 (as these are derived from a Perdew-Burke-Ernzerhof-like parametrization), could not yield improvements over available functionals when applied on a test set of systems designed to probe solvatochromic shifts. Moreover, as several different functionals yielded very similar errors to the simple local-density approximation (LDA), it is questionable whether it is worthwhile to go beyond the LDA in this context.     

Kinetic energy density study of some representative semilocal kinetic energy functionals

There is a number of explicit kinetic energy density functionals for noninteracting electron systems that are obtained in terms of the electron density and its derivatives. These semilocal functionals have been widely used in the literature. In this work, we present a comparative study of the kinetic energy density of these semilocal functionals, stressing the importance of the local behavior to assess the quality of the functionals. We propose a quality factor that measures the local differences between the usual orbital-based kinetic energy density distributions and the approximated ones, allowing us to ensure if the good results obtained for the total kinetic energies with these semilocal functionals are due to their correct local performance or to error cancellations. We have also included contributions coming from the Laplacian of the electron density to work with an infinite set of kinetic energy densities. For all but one of the functionals, we have found that their success in the evaluation of the total kinetic energy is due to global error cancellations, whereas the local behavior of their kinetic energy density becomes worse than that corresponding to the Thomas-Fermi functional.     

Magnetic exchange couplings evaluated with Rung 3.5 density functionals

Rung 3.5 exchange-correlation functionals are assessed for the calculation of magnetic exchange coupling parameters and atomic spin populations for a variety of inorganic and organic magnetic systems. Density functional theory calculations of exchange couplings sensitively depend on nonlocal contributions to the exchange-correlation functional. Semilocal functionals, Rungs 1-3 on "Jacob's Ladder" of density functional approximations, yield excessively delocalized electrons and overestimated absolute exchange couplings. Fourth-rung hybrid functionals admixing nonlocal exchange improve the results. We show that new "Rung 3.5" functionals give magnetic properties intermediate between semilocal and hybrid functionals, providing additional evidence that these functionals incorporate some desirable aspects of nonlocal exchange. Results for ferromagnetic complexes indicate areas for future improvement.     

Orbital-free density functional theory applied to NaAlH4

We present the application of orbital-free density functional theory (OF-DFT) to NaAlH(4), a potential hydrogen storage material, and related systems. Although the simple Al and NaH structures are reproduced reasonably well by OF-DFT, the approach fails for the more complex NaAlH(4) structure. Calculations on AlH(3) show that the failure to describe the Al-H interaction is related to the kinetic energy functionals used rather than the local pseudopotentials which are required within the OF-DFT approach. Thus, systems such as NaAlH(4) present a challenge which awaits the development of more reliable orbital-free kinetic energy functionals.     

Gradient expansion of the kinetic energy density functional: Local behavior of the kinetic energy density

Calculations with Hartree—Fock electron densities for the rare gas atoms He through Xe show that the gradient expansion for the kinetic energy functional, T[] = T₀[] + T₂[] + T₄[] + … = ∫t() dτ, approximates the kinetic energy by averaging over the shell structure present in the true local kinetic energy density, t(), and that the accuracy of the gradient expansion improves with increasing atomic number. Components of t(), t₀(), t₂() and t₄(), are exhibited and discussed. The defined function t() is everywhere positive.     

Binding energy curves from nonempirical density functionals. I. Covalent bonds in closed-shell and radical molecules

Binding or potential energy curves have been calculated for the ground-state diatomics H(2)(+), He(2)(+), LiH(+), H(2), N(2), and C(2), for the transition state H(3), and for the triplet first excited state of H(2) using the nonempirical density functionals from the first three rungs of a ladder of approximations: the local spin density (LSD) approximation, the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), and the Tao-Perdew-Staroverov-Scuseria (TPSS) meta GGA. Good binding energy curves in agreement with coupled cluster or configuration interaction calculations are found from the PBE GGA and especially from the TPSS meta GGA. Expected exceptions are the symmetric radicals H(2)(+) and He(2)(+), where the functionals suffer from self-interaction error, and the exotically bonded C(2). Although the energy barrier for the reaction H(2) + H --> H + H(2) is better in PBE than in TPSS, the transition state H(3) is a more properly positioned and curved saddle point of the energy surface in TPSS. The triplet first excited state of H(2) obeys the Aufbau principle and thus is one of the exceptional excited states that are computable in principle from the ground-state functional. The PBE GGA and TPSS meta GGA are useful not only for chemical applications but also for the construction of higher-rung nonempirical functionals that can further improve the binding energy curves.     

Effective homogeneity of the exchange-correlation and non-interacting kinetic energy functionals under density scaling

Correlated electron densities, experimental ionisation potentials, and experimental electron affinities are used to investigate the homogeneity of the exchange-correlation and non-interacting kinetic energy functionals of Kohn-Sham density functional theory under density scaling. Results are presented for atoms and small molecules, paying attention to the influence of the integer discontinuity and the choice of the electron affinity. For the exchange-correlation functional, effective homogeneities are highly system-dependent on either side of the integer discontinuity. By contrast, the average homogeneity-associated with the potential that averages over the discontinuity-is generally close to 4/3 when the discontinuity is computed using positive affinities for systems that do bind an excess electron and negative affinities for those that do not. The proximity to 4/3 becomes increasingly pronounced with increasing atomic number. Evaluating the discontinuity using a zero affinity in systems that do not bind an excess electron instead leads to effective homogeneities on the electron abundant side that are close to 4/3. For the non-interacting kinetic energy functional, the effective homogeneities are less system-dependent and the effect of the integer discontinuity is less pronounced. Average values are uniformly below 5/3. The study provides information that may aid the development of improved exchange-correlation and non-interacting kinetic energy functionals.     

Kinetic energy density for orbital‐free density functional calculations by axiomatic approach

An axiomatic approach is herein used to determine the physically acceptable forms for general D‐dimensional kinetic energy density functionals (KEDF). The resulted expansion captures most of the known forms of one‐point KEDFs. By statistically training the KEDF forms on a model problem of noninteracting kinetic energy in 1D (six terms only), the mean relative accuracy for 1000 randomly generated potentials is found to be better than the standard KEDF by several orders of magnitudes. The accuracy improves with the number of occupied states and was found to be better than for a system with four occupied states. Furthermore, we show that free fitting of the coefficients associated with known KEDFs approaches the exactly analytic values. The presented approach can open a new route to search for physically acceptable kinetic energy density functionals and provide an essential step toward more accurate large‐scale orbital free density functional theory calculations.     

Fluctuation-dissipation theorem density-functional theory

Using the fluctuation-dissipation theorem (FDT) in the context of density-functional theory (DFT), one can derive an exact expression for the ground-state correlation energy in terms of the frequency-dependent density response function. When combined with time-dependent density-functional theory, a new class of density functionals results that use approximations to the exchange-correlation kernel fxc as input. This FDT-DFT scheme holds promise to solve two of the most distressing problems of conventional Kohn-Sham DFT: (i) It leads to correlation energy functionals compatible with exact exchange, and (ii) it naturally includes dispersion. The price is a moderately expensive O(N6) scaling of computational cost and a slower basis set convergence. These general features of FDT-DFT have all been recognized previously. In this paper, we present the first benchmark results for a set of molecules using FDT-DFT beyond the random-phase approximation (RPA)-that is, the first such results with fxc not equal to 0. We show that kernels derived from the adiabatic local-density approximation and other semilocal functionals suffer from an "ultraviolet catastrophe," producing a pair density that diverges at small interparticle distance. Nevertheless, dispersion interactions can be treated accurately if hybrid functionals are employed, as is demonstrated for He2 and HeNe. We outline constraints that future approximations to fxc should satisfy and discuss the prospects of FDT-DFT.     

Systematic studies on the computation of nuclear magnetic resonance shielding constants and chemical shifts: the density functional models

We present a systematic density functional investigation on the prediction of the 13C, 15N, 17O, and 19F NMR properties of 23 molecules with 21 density functionals. Extensive comparisons are made for both 13C magnetic shieldings and chemical shifts with respect to the gas phase experimental data and the best CCSD(T) results. We find that the OPBE and OPW91 exchange-correlation functionals perform significantly better than some popular functionals such as B3LYP and PBE1PBE, even surpassing, in many cases, the standard wavefunction-based method MP2. Further analysis has been performed to explore the individual role played by various exchange and correlation functionals. We find that the B88 and PBE exchange functionals have a too strong tendency of deshielding, leading to too deshielded magnetic shielding constants; whereas the OPTX exchange functional performs remarkably well. We claim that the main source of error arises from the exchange functional, but correlation functional also makes important contribution. We find that the correlation functionals may be grouped into two classes. class A, such as LYP and B98, leads to deshielded NMR values, deteriorating the overall performance; whereas class B, such as PW91 and PBE, generally increases the absolute shieldings, which complements the exchange functionals, leading to improved results in the calculation of NMR data.     

Density functionals for exchange and correlation energies: Exact conditions and comparison of approximations

A set of exact conditions is compiled for the purpose of developing and testing approximations for the exchange-correlation energy as a functional of the electron density. Special emphasis is placed upon recently developed density-scaling relationships. Commonly used generalized gradient approximations are compared against several of these conditions. A direct tabular comparison of these functionals (not of calculated properties) with one another is also made. © 1994 John Wiley & Sons, Inc.     

Gauge invariant calculations of nuclear magnetic shielding constants using the continuous transformation of the origin of the current density approach. II. Density functional and coupled cluster theory

The quantum mechanical current density induced in a molecule by an external magnetic field is invariant to translations of the coordinate system. This fundamental symmetry is exploited to formally annihilate the diamagnetic contribution to the current density via the approach of "continuous transformation of the origin of the current density-diamagnetic zero" (CTOCD-DZ). The relationships obtained by this method for the magnetic shielding at the nuclei are intrinsically independent of the origin of the coordinate system for any approximate computational scheme relying on the algebraic approximation. The authors report for the first time an extended series of origin-independent estimates of nuclear magnetic shielding constants using the CTOCD-DZ approach at the level of density functional theory (DFT) with four different types of functionals and unrelaxed coupled cluster singles and doubles linear response (CCSD-LR) theory. The results obtained indicate that in the case of DFT the procedure employed is competitive with currently adopted computational methods allowing for basis sets of gauge-including atomic orbitals, whereas larger differences between CTOCD-DZ and common origin CCSD-LR results are observed due to the incomplete fulfillment of hypervirial relations in standard CCSD-LR theory. It was found furthermore that the unrelaxed CCSD-LR calculations predict larger correlation corrections for the shielding constants of almost all nonhydrogen atoms in their set of molecules than the usual relaxed energy derivative CCSD calculations. Finally the results confirm the excellent performance of Keal and Tozer's third functional, in particular, for the multiply bonded systems with a lot of electron correlation, but find also that the simple local density functional gives even better results for the few singly bonded molecules in their study where correlation effects are small.     

Generalized nonlocal kinetic energy density functionals based on the von Weizsäcker functional

We generalize the ideas behind the procedure for the construction of kinetic energy density functionals with a nonlocal term based on the structure of the von Weizs?cker functional, and present several types of nonlocal terms. In all cases, the functionals are constructed such that they reproduce the linear response function of the homogeneous electron gas. These functionals are designed by rewriting the von Weizs?cker functional with the help of a parameter β that determines the power of the electron density in the expression, a strategy we have previously used in the generalization of Thomas-Fermi nonlocal functionals. Benchmark calculations in localized systems have been performed with these functionals to test both their relative errors and the quality of their local behavior. We have obtained competitive results when compared to semilocal and previous nonlocal functionals, the generalized nonlocal von Weizs?cker functionals giving very good results for the total kinetic energies and improving the local behavior of the kinetic energy density. In addition, all the functionals discussed in this paper, when using an adequate reference density, can be evaluated as a single integral in momentum space, resulting in a quasilinear scaling for the computational cost.     

Standard grids for high‐precision integration of modern density functionals: SG‐2 and SG‐3

Density‐functional approximations developed in the past decade necessitate the use of quadrature grids that are far more dense than those required to integrate older generations of functionals. This category of difficult‐to‐integrate functionals includes meta‐generalized gradient approximations, which depend on orbital gradients and/or the Laplacian of the density, as well as functionals based on B97 and the popular “Minnesota” class of functionals, each of which contain complicated and/or oscillatory expressions for the exchange inhomogeneity factor. Following a strategy introduced previously by Gill and co‐workers to develop the relatively sparse “SG‐0” and “SG‐1” standard quadrature grids, we introduce two higher‐quality grids that we designate SG‐2 and SG‐3, obtained by systematically “pruning” medium‐ and high‐quality atom‐centered grids. The pruning procedure affords computational speedups approaching a factor of two for hybrid functionals applied to systems of atoms, without significant loss of accuracy. The grid dependence of several popular density functionals is characterized for various properties. © 2017 Wiley Periodicals, Inc.     

Failure of the Weizs?cker kinetic energy functional for one-, two-, and three-electron distribution functions

Further progress in pair-density functional theory (sometimes called 2-DFT) hinges on the development of computationally facile and quantitatively accurate models for the kinetic energy functional. In this paper we perform computational tests for two of the simplest models, the generalized Weizs?cker kinetic energy functional and its spin-resolved extension. Both of these models perform very poorly for atoms. The higher-order Weizs?cker functionals (based on the three-electron distribution function) perform better, but are still not successful. This suggests that an alternative approach for designing kinetic energy functionals of the pair density is needed.     

Assessment of the pairwise additive approximation and evaluation of many-body terms for water clusters

We have tested the ability of four commonly used density functionals (three of which are semilocal and one of which is nonlocal) to outperform accurate pairwise additive approximations in the prediction of binding energies for a series of water clusters ranging in size from dimer to pentamer. Comparison to results obtained with the Weizmann-1 (W1) level of wave function theory shows that while all density functionals are capable of outperforming the accurate pairwise data, the choice of basis set used is crucial to the performance of the method, and if a poor choice of basis set is made the errors obtained with density functional theory (DFT) can be larger than those obtained with the simple pairwise approximation. We have also compared the binding energies and many-body terms determined with DFT to those obtained with W1, and have found that all semilocal functionals have significant errors in the many-body components of the full interactions energy. Despite this limitation, however, we have found that, of the four functionals tested, PBE1W/MG3S is the most accurate for predicting the binding energies of the clusters.     

Performance of conventional and range-separated hybrid density functionals in calculations of electronic circular dichroism spectra of transition metal complexes

A number of density functionals, including 'pure' (nonhybrid) functionals, global hybrids, and range-separated hybrids, were used to calculate the electronic circular dichroism (CD) spectra of 10 tris-bidentate transition metal complexes. The results are compared to one another and to experimental CD spectra, in an effort to illustrate the shortcomings of particular approximations in time-dependent density functional theory (TDDFT). The use of an origin invariant formalism to calculate magnetic transition dipole moments with the help of gauge-including atomic orbitals (GIAOs) is also investigated. With valence basis sets of moderate flexibility, good agreement between GIAO results and rotatory strengths calculated from the dipole-velocity representation is obtained for selected test cases. Empirically broadened vertical CD spectra calculated with the global hybrid functionals B3LYP and PBE0 are found to agree overall the best with experimental CD spectra.